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Introduc)on to Bayesian methods Lecture 10

David Sontag New York University

Slides adapted from Luke Zettlemoyer, Carlos Guestrin, Dan Klein, and Vibhav Gogate

Bayesian learning

•  Bayesian learning uses probability to model data and quan+fy uncertainty of predic;ons – Facilitates incorpora;on of prior knowledge – Gives op;mal predic;ons

•  Allows for decision-‐theore;c reasoning

Your first consul;ng job •  A billionaire from the suburbs of ManhaFan asks you a ques;on: – He says: I have thumbtack, if I flip it, what’s the probability it will fall with the nail up?

–  You say: Please flip it a few ;mes:

–  You say: The probability is: •  P(heads) = 3/5

– He says: Why??? –  You say: Because…

Outline of lecture

•  Review of probability •  Maximum likelihood es;ma;on

2 examples of Bayesian classifiers:

•  Naïve Bayes •  Logis;c regression

Random Variables

•  A random variable is some aspect of the world about which we (may) have uncertainty –  R = Is it raining? –  D = How long will it take to drive to work? –  L = Where am I?

•  We denote random variables with capital letters

•  Random variables have domains –  R in {true, false} (sometimes write as {+r, ¬r}) –  D in [0, ∞) –  L in possible locations, maybe {(0,0), (0,1), …}

Probability Distributions •  Discrete random variables have distributions

•  A discrete distribution is a TABLE of probabilities of values

•  The probability of a state (lower case) is a single number

•  Must have:

warm 0.5

cold 0.5

sun 0.6

rain 0.1

fog 0.3

meteor 0.0

Joint Distributions •  A joint distribution over a set of random variables:

specifies a real number for each assignment:

–  How many assignments if n variables with domain sizes d?

–  Must obey:

•  For all but the smallest distributions, impractical to write out or estimate –  Instead, we make additional assumptions about the distribution

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

Marginal Distributions

•  Marginal distributions are sub-tables which eliminate variables

•  Marginalization (summing out): Combine collapsed rows by adding

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

hot 0.5

cold 0.5

sun 0.6

rain 0.4

P (t) =X

P (t, w)

P (w) =X

P (t, w)

Conditional Probabilities •  A simple relation between joint and conditional probabilities

–  In fact, this is taken as the definition of a conditional probability

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

Conditional Distributions •  Conditional distributions are probability distributions over

some variables given fixed values of others

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

sun 0.8

rain 0.2

sun 0.4

rain 0.6

Conditional Distributions Joint Distribution

The Product Rule

•  Sometimes have conditional distributions but want the joint

•  Example:

sun 0.8

rain 0.2

wet sun 0.1

dry sun 0.9

wet rain 0.7

dry rain 0.3

wet sun 0.08

dry sun 0.72

wet rain 0.14

dry rain 0.06

Bayes’ Rule

•  Two ways to factor a joint distribution over two variables:

•  Dividing, we get:

•  Why is this at all helpful? –  Let’s us build one conditional from its reverse –  Often one conditional is tricky but the other one is simple –  Foundation of many practical systems (e.g. ASR, MT)

•  In the running for most important ML equation!

Returning to thumbtack example…

•  P(Heads) = θ, P(Tails) = 1-‐θ

•  Flips are i.i.d.: –  Independent events –  Iden;cally distributed according to Bernoulli distribu;on

•  Sequence D of αH Heads and αT Tails

D={xi | i=1…n}, P(D | θ ) = ΠiP(xi | θ )

Called the “likelihood” of the data under the model

Maximum Likelihood Es;ma;on •  Data: Observed set D of αH Heads and αT Tails •  Hypothesis: Bernoulli distribu;on •  Learning: finding θ is an op;miza;on problem

– What’s the objec;ve func;on?

•  MLE: Choose θ to maximize probability of D

Your first parameter learning algorithm

•  Set deriva;ve to zero, and solve!

Brief Article

The Author

January 11, 2012

ˆ⇥ = argmax

⇥lnP (D | ⇥)

ln ⇥�H

d⇥lnP (D | ⇥) =

ln ⇥�H(1� ⇥)�T

�H ln ⇥ + �T ln(1� ⇥)]

Brief Article

The Author

January 11, 2012

ˆ⇥ = argmax

⇥lnP (D | ⇥)

ln ⇥�H

d⇥lnP (D | ⇥) =

ln ⇥�H(1� ⇥)�T

�H ln ⇥ + �T ln(1� ⇥)]

Brief Article

The Author

January 11, 2012

ˆ⇥ = argmax

⇥lnP (D | ⇥)

ln ⇥�H

d⇥lnP (D | ⇥) =

ln ⇥�H(1� ⇥)�T

�H ln ⇥ + �T ln(1� ⇥)]

= �Hd

d⇥ln ⇥ + �T

d⇥ln(1� ⇥) =

⇥� �T

1� ⇥= 0

Brief Article

The Author

January 11, 2012

ˆ⇥ = argmax

⇥lnP (D | ⇥)

ln ⇥�H

d⇥lnP (D | ⇥) =

ln ⇥�H(1� ⇥)�T

�H ln ⇥ + �T ln(1� ⇥)]

= �Hd

d⇥ln ⇥ + �T

d⇥ln(1� ⇥) =

⇥� �T

1� ⇥= 0

Brief Article

The Author

January 11, 2012

ˆ� = argmax

✓lnP (D | �)

ln �↵H

d�lnP (D | �) =

d�ln �↵H

(1� �)↵T

Brief Article

The Author

January 11, 2012

ˆ⇥ = argmax

⇥lnP (D | ⇥)

ln ⇥�H

d⇥lnP (D | ⇥) =

ln ⇥�H(1� ⇥)�T

�H ln ⇥ + �T ln(1� ⇥)]

= �Hd

d⇥ln ⇥ + �T

d⇥ln(1� ⇥) =

⇥� �T

1� ⇥= 0

L(✓;D) = lnP (D|✓)

What if I have prior beliefs? •  Billionaire says: Wait, I know that the thumbtack is “close” to 50-‐50. What can you do for me now?

•  You say: I can learn it the Bayesian way… •  Rather than es;ma;ng a single θ, we obtain a distribu;on over possible values of θ

Observe flips e.g.: {tails, tails}

In the beginning

Pr(✓)

After observations

Pr(✓ | D)

Bayesian Learning

•  Use Bayes’ rule!

•  Or equivalently:

•  For uniform priors, this reduces to

maximum likelihood estimation!

Normalization

Data Likelihood

Posterior

Brief Article

The Author

January 11, 2012

ˆ⌅ = argmax

⇤lnP (D | ⌅)

ln ⌅�H

d⌅lnP (D | ⌅) =

ln ⌅�H(1� ⌅)�T

�H ln ⌅ + �T ln(1� ⌅)]

= �Hd

d⌅ln ⌅ + �T

d⌅ln(1� ⌅) =

⌅� �T

1� ⌅= 0

⇥ ⇤ 2e�2N⇥2 ⇤ P (mistake)

ln ⇥ ⇤ ln 2� 2N⇤2

N ⇤ ln(2/⇥)

N ⇤ ln(2/0.05)2⇥ 0.12 ⌅ 3.8

0.02= 190

P (⌅) ⇧ 1

Brief Article

The Author

January 11, 2012

ˆ⌅ = argmax

⇤lnP (D | ⌅)

ln ⌅�H

d⌅lnP (D | ⌅) =

ln ⌅�H(1� ⌅)�T

�H ln ⌅ + �T ln(1� ⌅)]

= �Hd

d⌅ln ⌅ + �T

d⌅ln(1� ⌅) =

⌅� �T

1� ⌅= 0

⇥ ⇤ 2e�2N⇥2 ⇤ P (mistake)

ln ⇥ ⇤ ln 2� 2N⇤2

N ⇤ ln(2/⇥)

N ⇤ ln(2/0.05)2⇥ 0.12 ⌅ 3.8

0.02= 190

P (⌅) ⇧ 1

P (⌅ | D) ⇧ P (D | ⌅)

Bayesian Learning for Thumbtacks

Likelihood:

•  What should the prior be? –  Represent expert knowledge –  Simple posterior form

•  For binary variables, commonly used prior is the Beta distribu;on:

Beta prior distribu;on – P(θ)

•  Since the Beta distribu;on is conjugate to the Bernoulli distribu;on, the posterior distribu;on has a par;cularly simple form:

Brief Article

The Author

January 11, 2012

ˆ⌅ = argmax

⌅lnP (D | ⌅)

ln ⌅�H

d⌅lnP (D | ⌅) =

ln ⌅�H(1� ⌅)�T

�H ln ⌅ + �T ln(1� ⌅)]

= �Hd

d⌅ln ⌅ + �T

d⌅ln(1� ⌅) =

⌅� �T

1� ⌅= 0

⇥ ⇤ 2e�2N⇤2 ⇤ P (mistake)

ln ⇥ ⇤ ln 2� 2N⇤2

N ⇤ ln(2/⇥)

N ⇤ ln(2/0.05)2⇥ 0.12 ⌅ 3.8

0.02= 190

P (⌅) ⇧ 1

P (⌅ | D) ⇧ P (D | ⌅)

P (⌅ | D) ⇧ ⌅�H(1� ⌅)�T ⌅⇥H�1

(1� ⌅)⇥T�1

Brief Article

The Author

January 11, 2012

ˆ⌅ = argmax

⌅lnP (D | ⌅)

ln ⌅�H

d⌅lnP (D | ⌅) =

ln ⌅�H(1� ⌅)�T

�H ln ⌅ + �T ln(1� ⌅)]

= �Hd

d⌅ln ⌅ + �T

d⌅ln(1� ⌅) =

⌅� �T

1� ⌅= 0

⇥ ⇤ 2e�2N⇤2 ⇤ P (mistake)

ln ⇥ ⇤ ln 2� 2N⇤2

N ⇤ ln(2/⇥)

N ⇤ ln(2/0.05)2⇥ 0.12 ⌅ 3.8

0.02= 190

P (⌅) ⇧ 1

P (⌅ | D) ⇧ P (D | ⌅)

P (⌅ | D) ⇧ ⌅�H(1� ⌅)�T ⌅⇥H�1

(1� ⌅)⇥T�1= ⌅�H+⇥H�1

(1� ⌅)�T+⇥t+1

Brief Article

The Author

January 11, 2012

ˆ⇧ = argmax

⌅lnP (D | ⇧)

ln ⇧�H

d⇧lnP (D | ⇧) =

ln ⇧�H(1� ⇧)�T

�H ln ⇧ + �T ln(1� ⇧)]

= �Hd

d⇧ln ⇧ + �T

d⇧ln(1� ⇧) =

⇧� �T

1� ⇧= 0

⇤ ⇤ 2e�2N⇤2 ⇤ P (mistake)

ln ⇤ ⇤ ln 2� 2N⌅2

N ⇤ ln(2/⇤)

N ⇤ ln(2/0.05)2⇥ 0.12 ⌅ 3.8

0.02= 190

P (⇧) ⇧ 1

P (⇧ | D) ⇧ P (D | ⇧)

P (⇧ | D) ⇧ ⇧�H(1�⇧)�T ⇧⇥H�1

(1�⇧)⇥T�1= ⇧�H+⇥H�1

(1�⇧)�T+⇥t+1= Beta(�H+⇥H , �T+⇥T )

•  We now have a distribu)on over parameters •  For any specific f, a func;on of interest, compute the

expected value of f:

•  Integral is oien hard to compute •  As more data is observed, prior is more concentrated

•  MAP (Maximum a posteriori approxima;on): use most likely parameter to approximate the expecta;on

Using Bayesian inference for predic;on

What about con;nuous variables?

•  Billionaire says: If I am measuring a con;nuous variable, what can you do for me?

•  You say: Let me tell you about Gaussians…

Some proper;es of Gaussians

•  Affine transforma;on (mul;plying by scalar and adding a constant) are Gaussian –  X ~ N(µ,σ2) –  Y = aX + b Y ~ N(aµ+b,a2σ2)

•  Sum of Gaussians is Gaussian –  X ~ N(µX,σ2

X) –  Y ~ N(µY,σ2

Y) –  Z = X+Y Z ~ N(µX+µY, σ2

X+σ2Y)

•  Easy to differen;ate, as we will see soon!

Learning a Gaussian

•  Collect a bunch of data – Hopefully, i.i.d. samples

– e.g., exam scores

•  Learn parameters – µ (“mean”) – σ (“variance”)

xi i =

Exam Score

… …

MLE for Gaussian:

•  Prob. of i.i.d. samples D={x1,…,xN}:

•  Log-likelihood of data:

µMLE , �MLE = argmaxµ,�

P (D | µ,�)

Your second learning algorithm: MLE for mean of a Gaussian

•  What’s MLE for mean?

P (D | µ,�)

= �NX

(xi � µ)

�2= 0

P (D | µ,�)

= �NX

(xi � µ)

�2= 0

P (D | µ,�)

= �NX

(xi � µ)

�2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

�3= 0

MLE for variance •  Again, set deriva;ve to zero:

P (D | µ,�)

= �NX

(xi � µ)

�2= 0

P (D | µ,�)

= �NX

(xi � µ)

�2= 0

= �N

(xi � µ)2

�3= 0

Learning Gaussian parameters

•  MLE:

•  MLE for the variance of a Gaussian is biased – Expected result of es;ma;on is not true parameter! – Unbiased variance es;mator:

Bayesian learning of Gaussian parameters

•  Conjugate priors – Mean: Gaussian prior

– Variance: Wishart Distribu;on

•  Prior for mean:

Outline of lecture

•  Naïve Bayes •  Logis;c regression

Bayesian Classifica;on

•  Problem statement: – Given features X1,X2,…,Xn – Predict a label Y

[Next several slides adapted from: Vibhav Gogate, Jonathan Huang, Luke Zettlemoyer, Carlos

Guestrin, and Dan Weld]

Example Applica;on

•  Digit Recogni)on

•  X1,…,Xn ∈ {0,1} (Black vs. White pixels)

•  Y ∈ {0,1,2,3,4,5,6,7,8,9}

Classifier 5

The Bayes Classifier

•  If we had the joint distribu;on on X1,…,Xn and Y, could predict using:

–  (for example: what is the probability that the image represents a 5 given its pixels?)

•  So … How do we compute that?

•  Use Bayes Rule!

•  Why did this help? Well, we think that we might be able to specify how features are “generated” by the class label

Normalization Constant

Likelihood Prior

•  Let’s expand this for our digit recogni;on task:

•  To classify, we’ll simply compute these probabili;es, one per class, and predict based on which one is largest

Model Parameters

•  How many parameters are required to specify the likelihood, P(X1,…,Xn|Y) ?

–  (Supposing that each image is 30x30 pixels)

•  The problem with explicitly modeling P(X1,…,Xn|Y) is that there are usually way too many parameters:

– We’ll run out of space

– We’ll run out of ;me –  And we’ll need tons of training data (which is usually not available)

Naïve Bayes •  Naïve Bayes assump;on:

–  Features are independent given class:

– More generally:

•  How many parameters now? •  Suppose X is composed of n binary features

The Naïve Bayes Classifier •  Given:

–  Prior P(Y) –  n condi;onally independent features X given the class Y

–  For each Xi, we have likelihood P(Xi|Y)

•  Decision rule:

If certain assumption holds, NB is optimal classifier! (they typically don’t)

X1 Xn X2

A Digit Recognizer

•  Input: pixel grids

•  Output: a digit 0-9

Are the naïve Bayes assump;ons realis;c here?

What has to be learned?

1 0.01

2 0.05

3 0.05

4 0.30

5 0.80

6 0.90

7 0.05

8 0.60

9 0.50

0 0.80

1 0.05

2 0.01

3 0.90

4 0.80

5 0.90

6 0.90

7 0.25

8 0.85

9 0.60

0 0.80

MLE for the parameters of NB •  Given dataset

–  Count(A=a,B=b) Ã number of examples where A=a and B=b

•  MLE for discrete NB, simply: –  Prior:

–  Observa;on distribu;on:

µMLE ,⇥MLE = argmaxµ,�

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (Y = y) =Count(Y = y)Py0 Count(Y = y�)

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (Xi = x|Y = y) =Count(Xi = x, Y = y)Px0 Count(Xi = x�, Y = y)

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

MLE for the parameters of NB

•  Training amounts to, for each of the classes, averaging all of the examples together:

MAP es;ma;on for NB •  Given dataset

–  Count(A=a,B=b) Ã number of examples where A=a and B=b

•  MAP es;ma;on for discrete NB, simply: –  Prior:

–  Observa;on distribu;on:

•  Called “smoothing”. Corresponds to Dirichlet prior!

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

P (D | µ,⇥)

= �NX

(xi � µ)

⇥2= 0

= �NX

xi +Nµ = 0

= �N

(xi � µ)2

⇥3= 0

argmaxwln

⇥⇥2�

�[tj �P

iwihi(xj)]2

= argmaxw

�[tj �P

iwihi(xj)]2

= argminw

[tj �X

wihi(xj)]2

+ |X_i|*a

What about if there is missing data? •  One of the key strengths of Bayesian approaches is that

they can naturally handle missing data Missing data

  Suppose don’t have value for some attribute Xi

  applicant’s credit history unknown   some medical test not performed on patient   how to compute P(X1=x1 … Xj=? … Xd=xd | y)

  Easy with Naïve Bayes   ignore attribute in instance

where its value is missing   compute likelihood based on observed attributes   no need to “fill in” or explicitly model missing values   based on conditional independence between attributes

Missing data (2)

•  Ex: three coin tosses: Event = {X1=H, X2=?, X3=T} •  event = head, unknown (either head or tail), tail •  event = {H,H,T} + {H,T,T} •  P(event) = P(H,H,T) + P(H,T,T)

•  General case: Xj has missing value

Summary   Naïve Bayes classifier

-  explicitly handles class priors -  “normalizes” across observations: outliers comparable -  assumption: all dependence is “explained” by class label

  Continuous example -  unable to handle correlated data

  Discrete example -  fooled by repetitions -  must deal with zero-frequency problem

-  Pros: -  handles missing data -  good computational complexity -  incremental updates

Computational complexity

•  One of the fastest learning methods •  O(nd+cd) training time complexity

•  c … number of classes •  n … number of instances •  d … number of dimensions (attributes) •  both learning and prediction •  no hidden constants (number of iterations, etc.) •  testing: O(ndc)

•  O(dc) space complexity •  only decision trees are more compact

[Slide from Victor Lavrenko and Nigel Goddard]

pageMehryar Mohri - Introduction to Machine Learning

Naive Bayes = Linear Classifier

Theorem: assume that for all . Then, the Naive Bayes classifier is defined by

Proof: observe that for any ,

xi � {0, 1} i � [1, N ]

x ⇥� sgn(w · x + b),

i � [1, N ]

logPr[xi | +1]Pr[xi | �1]

logPr[xi = 1 | +1]Pr[xi = 1 | �1]

� logPr[xi = 0 | +1]Pr[xi = 0 | �1]

�xi+log

Pr[xi = 0 | +1]Pr[xi = 0 | �1]

wi = log Pr[xi=1|+1]Pr[xi=1|�1] � log Pr[xi=0|+1]

Pr[xi=0|�1]

b = log Pr[+1]Pr[�1] +

�Ni=1 log Pr[xi=0|+1]

Pr[xi=0|�1] .

[Slide from Mehyrar Mohri]

Outline of lecture

•  Naïve Bayes •  Logis)c regression

[Next several slides adapted from: Vibhav Gogate, Luke Zettlemoyer, Carlos Guestrin, and Dan Weld]

Logis;c Regression

  Learn P(Y|X) directly!  Assume a particular functional form

✬  Linear classifier? On one side we say P(Y=1|X)=1, and on the other P(Y=1|X)=0

✬ But, this is not differentiable (hard to learn)… doesn’t allow for label noise...

P(Y=1)=0

P(Y=1)=1

Logis;c Regression Logistic function (Sigmoid):

•  Learn P(Y|X) directly! •  Assume a particular

functional form

•  Sigmoid applied to a linear function of the data:

Features can be discrete or continuous!

Copyright

c⇥ 2010, Tom M. Mitchell. 7

where the superscript j refers to the jth training example, and where �(Y = yk) is1 if Y = yk and 0 otherwise. Note the role of � here is to select only those trainingexamples for which Y = yk.

The maximum likelihood estimator for ⇥2ik is

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

This maximum likelihood estimator is biased, so the minimum variance unbi-ased estimator (MVUE) is sometimes used instead. It is

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

3 Logistic RegressionLogistic Regression is an approach to learning functions of the form f : X ⇤Y , orP(Y |X) in the case where Y is discrete-valued, and X = ⌅X1 . . .Xn⇧ is any vectorcontaining discrete or continuous variables. In this section we will primarily con-sider the case where Y is a boolean variable, in order to simplify notation. In thefinal subsection we extend our treatment to the case where Y takes on any finitenumber of discrete values.

Logistic Regression assumes a parametric form for the distribution P(Y |X),then directly estimates its parameters from the training data. The parametricmodel assumed by Logistic Regression in the case where Y is boolean is:

P(Y = 1|X) =1

1+ exp(w0 +⇤ni=1 wiXi)

andP(Y = 0|X) =

exp(w0 +⇤ni=1 wiXi)

Notice that equation (17) follows directly from equation (16), because the sum ofthese two probabilities must equal 1.

One highly convenient property of this form for P(Y |X) is that it leads to asimple linear expression for classification. To classify any given X we generallywant to assign the value yk that maximizes P(Y = yk|X). Put another way, weassign the label Y = 0 if the following condition holds:

1 <P(Y = 0|X)P(Y = 1|X)

substituting from equations (16) and (17), this becomes

1 < exp(w0 +n

⇤i=1

Copyright

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

P(Y = 1|X) =1

andP(Y = 0|X) =

1 <P(Y = 0|X)P(Y = 1|X)

1 < exp(w0 +n

⇤i=1

1 + e�z

Logis;c Func;on in n Dimensions

-2 0 2 4 6-4-2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1x1x2

Sigmoid applied to a linear function of the data:

Features can be discrete or continuous!

Logis;c Regression: decision boundary

A Linear Classifier!

•  Prediction: Output the Y with highest P(Y|X) –  For binary Y, output Y=0 if

Copyright

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

P(Y = 1|X) =1

andP(Y = 0|X) =

1 <P(Y = 0|X)P(Y = 1|X)

1 < exp(w0 +n

⇤i=1

Copyright

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

P(Y = 1|X) =1

andP(Y = 0|X) =

1 <P(Y = 0|X)P(Y = 1|X)

1 < exp(w0 +n

⇤i=1

Copyright

c� 2010, Tom M. Mitchell. 8

−5 0 50

Y = 1/(1 + exp(−X))

Figure 1: Form of the logistic function. In Logistic Regression, P(Y |X) is as-sumed to follow this form.

and taking the natural log of both sides we have a linear classification rule thatassigns label Y = 0 if X satisfies

0 < w0 +n

⇥i=1

wiXi (18)

and assigns Y = 1 otherwise.Interestingly, the parametric form of P(Y |X) used by Logistic Regression is

precisely the form implied by the assumptions of a Gaussian Naive Bayes classi-fier. Therefore, we can view Logistic Regression as a closely related alternative toGNB, though the two can produce different results in many cases.

3.1 Form of P(Y |X) for Gaussian Naive Bayes ClassifierHere we derive the form of P(Y |X) entailed by the assumptions of a GaussianNaive Bayes (GNB) classifier, showing that it is precisely the form used by Logis-tic Regression and summarized in equations (16) and (17). In particular, considera GNB based on the following modeling assumptions:

• Y is boolean, governed by a Bernoulli distribution, with parameter � =P(Y = 1)

• X = ⇤X1 . . .Xn⌅, where each Xi is a continuous random variable

Copyright

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

P(Y = 1|X) =1

andP(Y = 0|X) =

1 <P(Y = 0|X)P(Y = 1|X)

1 < exp(w0 +n

⇤i=1

Copyright

⇥̂2ik =

1⇤ j �(Y j = yk) ⇤

i � µ̂ik)2�(Y j = yk) (14)

⇥̂2ik =

1(⇤ j �(Y j = yk))�1 ⇤

i � µ̂ik)2�(Y j = yk) (15)

P(Y = 1|X) =1

andP(Y = 0|X) =

1 <P(Y = 0|X)P(Y = 1|X)

1 < exp(w0 +n

⇤i=1

Genera;ve (Naïve Bayes) maximizes Data likelihood

Discrimina;ve (Logis;c Regr.) maximizes Condi)onal Data Likelihood

Focuses only on learning P(Y|X) -‐ all that maFers for classifica;on

Likelihood vs. Condi;onal Likelihood

Maximizing Condi;onal Log Likelihood

Bad news: no closed-form solution to maximize l(w)

Good news: l(w) is concave function of w!

No local maxima Concave functions easy to optimize

0 or 1!

Op;mizing concave func;on – Gradient ascent

•  Condi;onal likelihood for Logis;c Regression is concave !

Gradient:

Update rule: Learning rate, η>0

Maximize Condi;onal Log Likelihood: Gradient ascent

error =�

(ti � t̂i)2 =

⇤ti �

wkhk(xi)

P (Y = 1|X) ⇥ exp(w10 +�

w1iXi)

P (Y = 2|X) ⇥ exp(w20 +�

w2iXi)

P (Y = r|X) = 1�r�1�

P (Y = j|X)

yj lnew0+

Pi wiXi

1 + ew0+P

i wiXi+ (1� yj) ln

1 + ew0+P

i wiXi

�l(w)

�yj � P (Y j = 1|xj , w)

�l(w)

⇧�

�wyj(w0 +

wixji )�

�wln

⇤1 + exp(w0 +

wixji )

⌅⌃

⇧yjxj

i �xj

i exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

⇧yj �

exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

error =�

(ti � t̂i)2 =

⇤ti �

wkhk(xi)

P (Y = 1|X) ⇥ exp(w10 +�

w1iXi)

P (Y = 2|X) ⇥ exp(w20 +�

w2iXi)

P (Y = r|X) = 1�r�1�

P (Y = j|X)

yj lnew0+

Pi wiXi

1 + ew0+P

i wiXi

�l(w)

�yj � P (Y j = 1|xj , w)

�l(w)

⇧�

�wyj(w0 +

wixji )�

�wln

⇤1 + exp(w0 +

wixji )

⌅⌃

⇧yjxj

i �xj

i exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

⇧yj �

exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

error =�

(ti � t̂i)2 =

⇤ti �

wkhk(xi)

P (Y = 1|X) ⇥ exp(w10 +�

w1iXi)

P (Y = 2|X) ⇥ exp(w20 +�

w2iXi)

P (Y = r|X) = 1�r�1�

P (Y = j|X)

yj lnew0+

Pi wiXi

1 + ew0+P

i wiXi

⌅l(w)

⌅wi=

�yj � P (Y j = 1|xj , w)

⌅l(w)

⌅wi=

⇧⌅

⌅wyj(w0 +

wixji )�

⌅wln

⇤1 + exp(w0 +

wixji )

⌅⌃

⇧yjxj

i �xj

i exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

⇧yj �

exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

1 + e�ax

ln p(w) ⇥ ��

⌅ ln p(w)

⌅wi= ��wi

⇤2⇥

e� (x�µik)

2�2i

error =�

(ti � t̂i)2 =

⇤ti �

wkhk(xi)

P (Y = 1|X) ⇥ exp(w10 +�

w1iXi)

P (Y = 2|X) ⇥ exp(w20 +�

w2iXi)

P (Y = r|X) = 1�r�1�

P (Y = j|X)

yj lnew0+

Pi wiXi

1 + ew0+P

i wiXi

⌅l(w)

⌅wi=

�yj � P (Y j = 1|xj , w)

⌅l(w)

⌅wi=

⇧⌅

⌅wyj(w0 +

wixji )�

⌅wln

⇤1 + exp(w0 +

wixji )

⌅⌃

⇧yjxj

i �xj

i exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

⇧yj �

exp(w0 +⌥

i wixji )

1 + exp(w0 +⌥

i wixji )

1 + e�ax

ln p(w) ⇥ ��

⌅ ln p(w)

⌅wi= ��wi

⇤2⇥

e� (x�µik)

2�2i

error =⌥

(ti � t̂i)2 =

⇤ti �

wkhk(xi)

P (Y = 1|X) ⇥ exp(w10 +⌥

w1iXi)

P (Y = 2|X) ⇥ exp(w20 +⌥

w2iXi)

P (Y = r|X) = 1�r�1⌥

P (Y = j|X)

yj lnew0+

Pi wiXi

1 + ew0+P

i wiXi

�l(w)

�yj � P (Y j = 1|xj , w)

�l(w)

⇧�

�wyj(w0 +

wixji )�

�wln

⇤1 + exp(w0 +

wixji )

⌅⌃

Gradient Ascent for LR

Gradient ascent algorithm: (learning rate η > 0)

For i=1 to n: (iterate over features)

un;l “change” < ε Loop over training examples!

Naïve Bayes vs. Logis)c Regression

Genera)ve •  Assume func;onal form for

–  P(X|Y) assume cond indep –  P(Y) –  Est. params from train data

•  Gaussian NB for cont. features •  Bayes rule to calc. P(Y|X= x):

–  P(Y | X) ∝ P(X | Y) P(Y) •  Indirect computa;on

–  Can generate a sample of the data –  Can easily handle missing data

Discrimina)ve •  Assume func;onal form for

–  P(Y|X) no assump)ons

–  Est params from training data •  Handles discrete & cont features

•  Directly calculate P(Y|X=x) –  Can’t generate data sample

Learning: h:X Y X – features Y – target classes

Naïve Bayes vs. Logis;c Regression

•  Genera;ve vs. Discrimina;ve classifiers •  Asympto;c comparison

(# training examples infinity) –  when model correct

•  NB, Linear Discriminant Analysis (with class independent variances), and Logis;c Regression produce iden;cal classifiers

–  when model incorrect •  LR is less biased – does not assume condi;onal independence

– therefore LR expected to outperform NB

[Ng & Jordan, 2002]

Naïve Bayes vs. Logis;c Regression

•  Genera;ve vs. Discrimina;ve classifiers •  Non-‐asympto;c analysis

–  convergence rate of parameter es;mates, (n = # of a_ributes in X) •  Size of training data to get close to infinite data solu;on •  Naïve Bayes needs O(log n) samples •  Logis;c Regression needs O(n) samples

– Naïve Bayes converges more quickly to its (perhaps less helpful) asympto;c es;mates

[Ng & Jordan, 2002]

Some experiments from UCI data

Naïve bayes Logistic Regression

lecture10 - People | MIT CSAILpeople.csail.mit.edu/dsontag/courses/ml14/slides/lecture... · 2014....

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